Dispersive Equations: a Survey

These notes were written as a guideline for a short talk; hence, the references and the statements of the theorems are often not given in full details. The point of these notes is to summarize the different directions that the study of dispersive equations has taken in the last ten years. I would like to mention here a few recent textbooks that treat different parts of this subject. They could be used as supporting material and as a source of new and interesting open problems: The Nonlinear Schrödinger Equation by C. Sulem and P.-L. Sulem [28], Semilinear Schrödinger Equations by T. Cazenave [8], Global Solutions of Nonlinear Schrödinger Equations by J. Bourgain [3], and finally, the recent book of T. Tao Nonlinear Dispersive Equations: Local and Global Analysis [29]. We start with two important examples of dispersive equations. These equations were introduced as models of certain physical phenomena; for example, see [28]. Here, we are not interested in understanding their derivation, but rather in studying the quantitative and qualitative properties of their wave solutions. Examples of Dispersive Equations • Nonlinear Schrödinger equation: 1. i∂tu + ∆u + N(u,Du) = 0, where u : M × R→ C, M = Rn, Tn or other manifolds. • The KdV equation: 2. ∂tu + ∂xxxu + γu∂xu = 0, where u : M × R→ R, M = R,T, and γ ∈ R. Why are these equations called dispersive? • Dispersion: “Dispersive” means that the solutions of these equations are waves that spread out spatially as long as no boundary conditions are imposed. A more mathematically precise characterization is the following: Consider the general linear evolution equation 3. ∂tu + P (D)u = 0 x ∈ Rn, where P (D) is a linear differential operator of symbol φ(ξ). Then, if one takes the space-time Fourier transform, it follows that